Fourier Series Coefficient Calculator
Understanding Fourier Series
What is a Fourier Series?
A Fourier series represents periodic functions as sums of sinusoidal functions.
f(x) = a₀/2 + Σ(aₙcos(nωx) + bₙsin(nωx))
where:
- a₀/2 is the constant term
- aₙ are cosine coefficients
- bₙ are sine coefficients
- ω = 2π/T (angular frequency)
- T is the period
Fourier Coefficients
Constant Term (a₀)
a₀ = (2/T)∫f(x)dx
- Average value over period
- DC component
- Mean of function
Cosine Terms (aₙ)
aₙ = (2/T)∫f(x)cos(nωx)dx
- Even symmetry
- Phase information
- Amplitude scaling
Sine Terms (bₙ)
bₙ = (2/T)∫f(x)sin(nωx)dx
- Odd symmetry
- Frequency components
- Harmonic content
Properties and Applications
Convergence
- Uniform convergence
- Gibbs phenomenon
- Rate of convergence
- Dirichlet conditions
Signal Processing
- Frequency analysis
- Filtering
- Compression
- Modulation
Special Properties
- Parseval's theorem
- Orthogonality
- Symmetry relations
- Energy distribution
Advanced Topics
Complex Form
- Exponential series
- Complex coefficients
- Polar representation
Extensions
- Fourier transform
- Discrete series
- Wavelets